conic sections, parabola

What if the directrix is not parallel to an axis? Suppose it is y=x+2, and let the focus be ((4,0)
How do I derive the equation of the parabola? The distance from the parabola is perpendicular to the directrix.

What if the directrix is not parallel to an axis? Suppose it is y=x+2, and let the focus be ((4,0)
How do I derive the equation of the parabola? The distance from the parabola is perpendicular to the directrix.

1. Use the complete definition of the parabola:
The distance of an arbitrary point P of the parabola to the focus F is equal to the perpendicular distance to the directrix d (= blue lines)

2. The triangle QFP is isosceles and the line MP is the perpendicular bisector of QF (by the way: MP is simultaneously the tangent to the parabola)

3. As posted the directrix has the equation y = x+2 and the focus is F(4,0). The point Q is located on the directrix and has the coordinates Q(q, q+2)

4. Then M has the coordinates

5. The slope of QF is and therefore the perpendicular slope is

6. The equation of the perpendicular bisector of QF is:

which will yield:

7. The perpendicular line to directrix passing through Q has the equation:

8. Now calculate the coordinates of the point P which is the intersection between the lines of #6 and #7:

After a lot of transformation you'll get:

9. Plug in this value into the equation at #7 to calculate the y-coordinate of P: